## Linear Logic (1992)

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Citations: | 24 - 1 self |

### BibTeX

@MISC{Lincoln92linearlogic,

author = {Patrick Lincoln},

title = {Linear Logic},

year = {1992}

}

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### Abstract

this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may read the sequent \Delta ` \Gamma as asserting that the multiplicative conjunction of the formulas in \Delta together imply the multiplicative disjunction of the formulas in \Gamma. A sequent calculus proof rule consists of a set of hypothesis sequents, displayed above a horizontal line, and a single conclusion sequent, displayed below the line, as below: Hypothesis1 Hypothesis2 Conclusion 4 Connections to Other Logics

### Citations

457 |
The mathematics of Sentence Structure
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Citation Context ... a type system [1, 15, 39, 40, 25, 8, 29, 41]. Other applications include analyzing the control structure of logic programs [7], generalized logic programming [4, 16], and natural language processing =-=[23]-=-. A natural characterization of polynomial time computations can be given in a bounded version of linear logic [13] obtained by limiting reuse to specified bounds, i.e., by bounding the number of refe... |

331 | Logic programming with focusing proofs in linear logic
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Citation Context ... the unique features of linear logic for use as a type system to study computational complexity [13] and compiler optimization techniques [40, 8, 29, 41, 34, 25], as well as uses in logic programming =-=[16, 3, 4]-=-, natural language processing [24, 38], and concurrency [5, 30, 35]. These recent contributions are developing linear logic from a theoretical curiosity into a tool that already has practical use with... |

303 | D.: Logic programming in a fragment of intuitionistic linear logic
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- 1994
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Citation Context ...tions by concentrating on linear logic as a type system [1, 15, 39, 40, 25, 8, 29, 41]. Other applications include analyzing the control structure of logic programs [7], generalized logic programming =-=[4, 16]-=-, and natural language processing [23]. A natural characterization of polynomial time computations can be given in a bounded version of linear logic [13] obtained by limiting reuse to specified bounds... |

207 | Linear Objects: logical processes with built-in inheritance
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- 1990
(Show Context)
Citation Context ...tions by concentrating on linear logic as a type system [1, 15, 39, 40, 25, 8, 29, 41]. Other applications include analyzing the control structure of logic programs [7], generalized logic programming =-=[4, 16]-=-, and natural language processing [23]. A natural characterization of polynomial time computations can be given in a bounded version of linear logic [13] obtained by limiting reuse to specified bounds... |

163 |
Interaction nets
- Lafont
- 1990
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Citation Context ...sequence of Petri net rule applications that transform the token set fa; c; d; dg to fc; dg. This connection has been well-studied [5, 14, 30, 6, 9], and extended to cover other models of concurrency =-=[22, 2, 35]-=-. Linear logic has also been applied to several other areas of computer science. One key application of the resource-sensitive aspect of the logic was the development of a functional programming langu... |

125 |
The complexity of word problem for commutative semigroups and polynomial ideals
- MAYR, MEYER
- 1982
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Citation Context ... ? fragment of linear logic. An equivalent fragment is\Omega ; \Gammaffi; !; ?, which suffices for the encoding of Petri net reachability questions, as shown above, and thus is at least expspace-hard =-=[31]-=-. It is currently unknown if this multiplicative-exponential fragment of linear logic is decidable or not. 6.1 Undecidability of Propositional Linear Logic The full proof of undecidability is presente... |

93 |
Recursive unsolvability of Post’s problem of ‘tag’ and other topics in the theory of Turing machines
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Citation Context ... has a finite set of instructions of five kinds: Increment-A, Increment-B, Decrement-A, Decrement-B, and Fork. The Increment and Decrement instructions operate as they do in standard counter machines =-=[33]-=-. The Fork instruction causes a machine to split into two independent machines: from state hQ i ; A; Bi a machine taking the transition Q i ForkQ j ; Q k results in two machines, hQ j ; A; Bi and hQ k... |

91 |
The linear abstract machine
- Lafont
- 1988
(Show Context)
Citation Context ...sensitive aspect of the logic was the development of a functional programming language implementation in which garbage collection was replaced by explicit duplication operations based on linear logic =-=[21]-=-. More recent work has attempted to find a linear logical basis for many optimizations in (lazy) functional programming language implementations by concentrating on linear logic as a type system [1, 1... |

90 | Decision problems for propositional linear logic
- Lincoln, Mitchell, et al.
- 1992
(Show Context)
Citation Context ...inear Logic Although propositional linear logic was known to be very expressive, for some time it was thought to be decidable. However, propositional linear logic was recently shown to be undecidable =-=[26]-=-. Several other complexity results are given in [26], including the pspacecompleteness of propositional linear logic without ! or ?. A key open complexity problem is the decision problem for the\Omega... |

89 | Decidability of Reachability in Vector Addition Systems
- Kosaraju
- 1982
(Show Context)
Citation Context ... problem is in some question. An encoding of Petri net reachability problems in this fragment has been studied in [5], but although expspace-hard [31], Petri net reachability is known to be decidable =-=[20]-=-. It is not known how much more expressive this fragment of linear logic might be. The fourth fragment containing only the multiplicative connectives is NP-Complete [17, 18]. Connectives Linear Logic ... |

88 | Is there a use for linear logic
- Wadler
- 1991
(Show Context)
Citation Context ... [21]. More recent work has attempted to find a linear logical basis for many optimizations in (lazy) functional programming language implementations by concentrating on linear logic as a type system =-=[1, 15, 39, 40, 25, 8, 29, 41]-=-. Other applications include analyzing the control structure of logic programs [7], generalized logic programming [4, 16], and natural language processing [23]. A natural characterization of polynomia... |

85 |
Quantales, observational logic and process semantics
- ABRAMSKY, J
- 1993
(Show Context)
Citation Context ...sequence of Petri net rule applications that transform the token set fa; c; d; dg to fc; dg. This connection has been well-studied [5, 14, 30, 6, 9], and extended to cover other models of concurrency =-=[22, 2, 35]-=-. Linear logic has also been applied to several other areas of computer science. One key application of the resource-sensitive aspect of the logic was the development of a functional programming langu... |

71 |
Single-threaded polymorphic lambda calculus
- Guzman, Hudak
- 1990
(Show Context)
Citation Context ... [21]. More recent work has attempted to find a linear logical basis for many optimizations in (lazy) functional programming language implementations by concentrating on linear logic as a type system =-=[1, 15, 39, 40, 25, 8, 29, 41]-=-. Other applications include analyzing the control structure of logic programs [7], generalized logic programming [4, 16], and natural language processing [23]. A natural characterization of polynomia... |

67 |
From Petri nets to linear logic
- Martí-Oliet, Meseguer
- 1991
(Show Context)
Citation Context ...d This sequent is provable in linear logic if and only if there is a sequence of Petri net rule applications that transform the token set fa; c; d; dg to fc; dg. This connection has been well-studied =-=[5, 14, 30, 6, 9]-=-, and extended to cover other models of concurrency [22, 2, 35]. Linear logic has also been applied to several other areas of computer science. One key application of the resource-sensitive aspect of ... |

62 |
Bounded linear logic: A modular approach to polynomial time computability
- Girard, Scedrov, et al.
- 1992
(Show Context)
Citation Context ...programs [7], generalized logic programming [4, 16], and natural language processing [23]. A natural characterization of polynomial time computations can be given in a bounded version of linear logic =-=[13]-=- obtained by limiting reuse to specified bounds, i.e., by bounding the number of references to each datum in memory. We now turn our attention to some questions of a more theoretical nature. 4 6 Compl... |

53 | A Brief Guide to Linear Logic
- Scedrov
- 1993
(Show Context)
Citation Context ...ents an intuitive overview of linear logic, some recent theoretical results, and some interesting applications of linear logic to computer science. Other introductions to linear logic may be found in =-=[12, 36]-=-. 2 Linear Logic vs Classical and Intuitionistic Logics Linear logic differs from classical and intuitionistic logic in several fundamental ways. Classical logic may be viewed as if it deals with stat... |

44 | Operational aspects of linear lambda calculus
- Lincoln, Mitchell
- 1992
(Show Context)
Citation Context ... [21]. More recent work has attempted to find a linear logical basis for many optimizations in (lazy) functional programming language implementations by concentrating on linear logic as a type system =-=[1, 15, 39, 40, 25, 8, 29, 41]-=-. Other applications include analyzing the control structure of logic programs [7], generalized logic programming [4, 16], and natural language processing [23]. A natural characterization of polynomia... |

44 |
Lilac: A functional programming language based on linear logic
- Mackie
- 1994
(Show Context)
Citation Context |

31 |
Computational interpretations of linear logic. Theoretical Computer Science 111
- Abramsky
- 1993
(Show Context)
Citation Context |

31 |
Normal process representatives
- Gehlot, Gunter
- 1990
(Show Context)
Citation Context ...d This sequent is provable in linear logic if and only if there is a sequence of Petri net rule applications that transform the token set fa; c; d; dg to fc; dg. This connection has been well-studied =-=[5, 14, 30, 6, 9]-=-, and extended to cover other models of concurrency [22, 2, 35]. Linear logic has also been applied to several other areas of computer science. One key application of the resource-sensitive aspect of ... |

30 | Constant-Only Multiplicative Linear Logic is NP-Complete
- Lincoln, Winkler
- 1994
(Show Context)
Citation Context ...t does not use any propositions, that is, using only the constants 1 and ? and the connectives\Omega and \Gammaffi. Thus this multiplicative constant-only fragment of linear logic is also np-complete =-=[28]-=-. The full proof of np-completeness is presented in [17]. 6.3 Summary of Linear Logic Complexity The following table summarizes some of the results thus far achieved in the study of the complexity of ... |

29 | A decidable fragment of predicate calculus
- Ketonen
- 1984
(Show Context)
Citation Context ...ed: Weakening Left \Delta ` \Sigma \Delta; A ` \Sigma \Delta; A; A ` \Gamma \Delta; A ` \Gamma Contraction Left Direct and Affine logic share with linear logic the elimination of the contraction rule =-=[19]-=-: i.e.propositions cannot be arbitrarily copied. However, both of these logics allow weakening. Relevance and Pertinent logic disallow weakening, but allow contraction: i.e.propositions cannot be arbi... |

24 |
Nets as tensor theories
- Gunter, Gehlot
- 1989
(Show Context)
Citation Context ...d This sequent is provable in linear logic if and only if there is a sequence of Petri net rule applications that transform the token set fa; c; d; dg to fc; dg. This connection has been well-studied =-=[5, 14, 30, 6, 9]-=-, and extended to cover other models of concurrency [22, 2, 35]. Linear logic has also been applied to several other areas of computer science. One key application of the resource-sensitive aspect of ... |

24 |
programming in linear logic is NP-complete
- Horn
- 1992
(Show Context)
Citation Context ...em is np-hard by reduction from 3-Partition, a problem which requires a perfect partitioning of groups of objects in much the same way that linear logic requires a complete accounting of propositions =-=[17, 18]-=-. Somewhat surprisingly, there is an alternate encoding of 3-Partition in multiplicative linear logic that does not use any propositions, that is, using only the constants 1 and ? and the connectives\... |

24 | There’s no substitute for linear logic
- Wadler
- 1992
(Show Context)
Citation Context |

23 |
The multiplicative fragment of linear logic is NP-complete
- Kanovich
- 1991
(Show Context)
Citation Context ...em is np-hard by reduction from 3-Partition, a problem which requires a perfect partitioning of groups of objects in much the same way that linear logic requires a complete accounting of propositions =-=[17, 18]-=-. Somewhat surprisingly, there is an alternate encoding of 3-Partition in multiplicative linear logic that does not use any propositions, that is, using only the constants 1 and ? and the connectives\... |

22 | Event Spaces and Their Linear Logic
- Pratt
- 1991
(Show Context)
Citation Context ...sequence of Petri net rule applications that transform the token set fa; c; d; dg to fc; dg. This connection has been well-studied [5, 14, 30, 6, 9], and extended to cover other models of concurrency =-=[22, 2, 35]-=-. Linear logic has also been applied to several other areas of computer science. One key application of the resource-sensitive aspect of the logic was the development of a functional programming langu... |

21 |
A logic for concurrency
- Asperti
- 1987
(Show Context)
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19 |
The undecidability of entailment and relevant implication
- Urquhart
- 1984
(Show Context)
Citation Context ...rbitrarily discarded, but can be copied. Pertinent logic is decidable [32], but Relevance logic adds a distributivity axiom, which is absent from linear logic, which makes Relevance logic undecidable =-=[37]-=-. Linear logic disallows both weakening and contraction in general, although they are allowed for modal (! and ?) formulas. Linear logic arose partly out of a study of intuitionistic implication. Gira... |

17 |
Linear logic and interference control
- O’Hearn
- 1991
(Show Context)
Citation Context ...agment of linear logic. Current work is progressing to exploit the unique features of linear logic for use as a type system to study computational complexity [13] and compiler optimization techniques =-=[40, 8, 29, 41, 34, 25]-=-, as well as uses in logic programming [16, 3, 4], natural language processing [24, 38], and concurrency [5, 30, 35]. These recent contributions are developing linear logic from a theoretical curiosit... |

17 | Linearity and laziness
- Wakeling, Runciman
- 1991
(Show Context)
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15 |
A linear semantics for allowed logic programs
- Cerrito
- 1990
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Citation Context ...nal programming language implementations by concentrating on linear logic as a type system [1, 15, 39, 40, 25, 8, 29, 41]. Other applications include analyzing the control structure of logic programs =-=[7]-=-, generalized logic programming [4, 16], and natural language processing [23]. A natural characterization of polynomial time computations can be given in a bounded version of linear logic [13] obtaine... |

15 | The linear abstract machine. Theoretical Computer Science, 59(1– 2):157–180 - Lafont - 1988 |

14 |
From categorial grammar to bilinear logic
- Lambek
- 1993
(Show Context)
Citation Context ... use as a type system to study computational complexity [13] and compiler optimization techniques [40, 8, 29, 41, 34, 25], as well as uses in logic programming [16, 3, 4], natural language processing =-=[24, 38]-=-, and concurrency [5, 30, 35]. These recent contributions are developing linear logic from a theoretical curiosity into a tool that already has practical use within mainstream computer science. Refere... |

13 |
Implicative formulae in the “proofs as computations” analogy
- Asperti, Ferrari, et al.
- 1990
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8 | 1843, A system of Logic - S |

7 |
Collected Works. Edited by M.E
- Gentzen
- 1969
(Show Context)
Citation Context ...ors. One could add quantifiers to form first (or higher) order linear logic, but for this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen =-=[10]-=-, uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may read the sequent \... |

6 |
Linear ML
- Chirimar, Gunter, et al.
- 1992
(Show Context)
Citation Context |

5 |
Topics in Modal and Many-valued Logic
- Meyer
- 1966
(Show Context)
Citation Context ...hese logics allow weakening. Relevance and Pertinent logic disallow weakening, but allow contraction: i.e.propositions cannot be arbitrarily discarded, but can be copied. Pertinent logic is decidable =-=[32]-=-, but Relevance logic adds a distributivity axiom, which is absent from linear logic, which makes Relevance logic undecidable [37]. Linear logic disallows both weakening and contraction in general, al... |

3 |
La logique lin'eaire. Pour La Science, ' Edition Francaise de Scientific American
- Girard
- 1990
(Show Context)
Citation Context ...ents an intuitive overview of linear logic, some recent theoretical results, and some interesting applications of linear logic to computer science. Other introductions to linear logic may be found in =-=[12, 36]-=-. 2 Linear Logic vs Classical and Intuitionistic Logics Linear logic differs from classical and intuitionistic logic in several fundamental ways. Classical logic may be viewed as if it deals with stat... |

2 |
Language in Action
- VanBentham
- 1991
(Show Context)
Citation Context ... use as a type system to study computational complexity [13] and compiler optimization techniques [40, 8, 29, 41, 34, 25], as well as uses in logic programming [16, 3, 4], natural language processing =-=[24, 38]-=-, and concurrency [5, 30, 35]. These recent contributions are developing linear logic from a theoretical curiosity into a tool that already has practical use within mainstream computer science. Refere... |

1 | Is there a use for linear logic? To Appear ACM/IFIP PEPM - North-Holland - 1991 |